Maxwell Bloch System Research Articles

Rogue waves on the periodic background for a higher-order nonlinear Schrödinger–Maxwell–Bloch system

In this paper, we construct the rogue wave solutions on the background of the Jacobian elliptic functions for a higher-order nonlinear Schrödinger–Maxwell–Bloch system. The Jacobian elliptic function traveling wave solutions as the seed solutions are considered. Through the approach of the nonlinearization of the Lax pair and Darboux transformation method, the rogue waves and the line rogue waves on the Jacobian elliptic functions dn and cn background are obtained, respectively.

Patterns of rogue waves in the sharp-line Maxwell–Bloch system

The Maxwell–Bloch system describes light-matter interactions in a semi-infinitely long one dimensional two-level optical medium. Rogue wave patterns in the Maxwell–Bloch system under sharp-line limit are analytically studied. It is shown that when single internal parameter in bilinear expressions of rogue waves gets large, these waves would exhibit clear geometric patterns, which comprise fundamental (Peregrine) rogue waves arranged in shapes such as triangle, pentagon, heptagon and nonagon structures, with a possible lower-order rogue wave at the center. These rogue wave patterns are analytically determined from the root structure of the Yablonskii–Vorob’ev polynomial hierarchy through dilation, rotation, stretch and shear. It is also shown that when multiple internal parameters in the rogue wave solutions get large, new rogue wave patterns would arise, including heart-shaped structures, fan-shaped structures, and many others. Analytically, these patterns are determined by the root structure of the Adler–Moser polynomials through a linear transformation. Comparison between analytical predictions of these rogue patterns and true solutions shows excellent agreement.

On Maxwell-Bloch Systems with Inhomogeneous Broadening and One-sided Nonzero Background

On Maxwell-Bloch Systems with Inhomogeneous Broadening and One-sided Nonzero Background

Various solitons induced by relative phase in the nonlinear Schrödinger Maxwell-Bloch system.

We study the effect of relative phase on the characteristics of rogue waves and solitons described by rational solutions in the nonlinear Schrödinger Maxwell-Bloch system. We derived the rational rogue wave and soliton solutions with adjustable relative phase and present the parameter range of different types of rogue waves and solitons. Our findings show that the relative phase can alter the distribution of rational solitons and even change the type of rational solitons, leading to a rich array of rational soliton types by adjusting the relative phase. However, the relative phase does not affect the structure of the rogue wave, because the relative phase of the rogue wave changes during evolution. In particular, we confirm that the rational solitons with varying relative phases and the rogue waves at corresponding different evolution positions share the same distribution mode. This relationship holds true for rogue waves or breathers and their stable counterparts solitons or periodic waves in different nonlinear systems. The implications of our study are significant for exploring fundamental excitation elements in nonlinear systems.

Inhomogeneous reduced Maxwell-Bloch system in nonlinear optics: Darboux-transformation and solitonic issues

Studies on nonlinear optics are active, providing the applications in modulators, fiber lasers, optical sensors, etc. In this paper, we investigate an inhomogeneous reduced Maxwell-Bloch system, which could describe the ultra-short optical pulses through an inhomogeneous medium. N-fold Darboux transformation is constructed via an existing Lax pair, where N is a positive integer. Generalized Darboux transformation is obtained via a limit procedure. One- and two-soliton solutions are determined with the aid of the N-fold Darboux transformation. Double-pole soliton solutions are derived via the generalized Darboux transformation. With respect to the electric field, microscopic polarization, phase information, and atomic inversion related to the inhomogeneous medium, one soliton, interaction between the two solitons, and double-pole soliton are graphically depicted. Effects of the coefficients in that system on those solitons are shown via the trigonometric and linear functions.

Inverse scattering transforms for the nonlocal Hirota–Maxwell–Bloch system

The Hirota and Maxwell–Bloch (H-MB) system is a mathematical model that can be used to describe the wave propagation in an erbium-doped nonlinear fiber with higher-order dispersion. By extending the parameter ω in the H-MB system to the complex domain, all the nonlocal forms of the H-MB system, including the reverse-space-time, complex reverse-space, complex reverse-time, complex reverse-space-time H-MB systems, are found. Then the Riemann–Hilbert problem of the nonlocal H-MB system is established to analyze the corresponding inverse scattering problem and construct the corresponding soliton solutions. When N = 1, the singularity of the one-soliton solutions of each nonlocal H-MB system is analyzed. When N = 2, we take the nonlocal reverse-space-time H-MB system and the nonlocal complex reverse-time H-MB system as examples, showing that a two-soliton can also be regarded as a superposition of two single-soliton in the nonlocal cases as .

The integrability, equivalence and solutions of two kinds of integrable deformed fourth-order matrix NLS equations

Based on the higher-order restricted flows, the first type of integrable deformed fourth-order matrix NLS equations, that is, the fourth-order matrix NLS equations with self-consistent sources (FMNLSSCS), is derived. By virtue of the $${\bar{\partial }}$$ -dressing method, the second type of integrable deformed fourth-order matrix NLS equations called the fourth-order matrix NLS–Maxwell–Bloch system (FMNLS-MB) is presented. We prove the equivalence of the FMNLSSCS and the FMNLS-MB successfully. Furthermore, N-soliton solutions are explicitly obtained by means of the Cauchy matrix method starting from corresponding Sylvester equation.

Explicit solutions from Darboux transformation for the two-component nonlocal Hirota and Maxwell–Bloch system

Based on the two-component Hirota and Maxwell–Bloch (H-MB) system, which describes the propagation of ultra-short pulses in erbium-doped nonlinear optical fibers, we derive two kinds of two-component nonlocal H-MB (NH-MB) systems by choosing suitable symmetries, namely [Formula: see text]-symmetric two-component NH-MB system and reverse space-time two-component NH-MB system. Then the explicit solutions of these two systems are obtained by the Darboux transformation.

Degenerate soliton, breather and mixed solutions for a coupled Hirota–Maxwell–Bloch system in an erbium-doped fiber

Degenerate soliton, breather and mixed solutions for a coupled Hirota–Maxwell–Bloch system in an erbium-doped fiber

The N-soliton solution to the higher-order nonlinear Schrödinger–Maxwell–Bloch system via the Riemann–Hilbert approach

In this paper, we study the N-soliton solution of the high-order nonlinear Schrödinger–Maxwell–Bloch system via the Riemann–Hilbert approach. Based on a Lax pair of the system, we carry out a spectral analysis and construct the corresponding Riemann–Hilbert problem. By solving the Riemann–Hilbert problem without reflection, we obtain the N-soliton solution of the system, especially the explicit expression of the single-soliton solution and the expression of the two-soliton solution. The partial three-dimensional diagrams of single-soliton and two-soliton solutions are also given. By comparing the diagrams, we also discuss the relationship between the parameters of the system and the soliton direction.

Soliton solutions for Hirota–Maxwell–Bloch system and its nonlocal form

This paper studies the Hirota–Maxwell–Bloch (H–MB) system and its nonlocal form. Based on the Darboux Transformations (DTs), for H–MB system, we present general double breathers, what is more, we take appropriate modulation frequency and position parameters to investigate the generative mechanism of rogue wave sequences and different periodic breather sequences. For nonlocal Hirota–Maxwell–Bloch (NH–MB) system, we discuss symmetry preserving and broken soliton solutions under zero background. Besides, we present nine combinations of dark and antidark soliton solutions under continuous waves background when PT-symmetry is broken.

Rational Integrability of the Maxwell–Bloch System

The Maxwell–Bloch system arises naturally from laser models, which describes the dynamics of a two-state quantum system coupled to the basic electromagnetic resonator model. In this paper, we study its integrability and give a complete classification of the irreducible Darboux polynomials, which can help us find all polynomial and rational first integrals of the Maxwell–Bloch system. To this end, in the case that the loss rate of field is zero, we use the characteristic curve method and weight homogeneous polynomials to investigate the integrability of the Maxwell–Bloch system, and in the case that the loss rate of field is nonzero, we provide a linear scaling of variables to transform the Maxwell–Bloch system into the Lorenz system and obtain its integrability from some corresponding results of the Lorenz system. Our results show that although the Maxwell–Bloch system has chaotic behaviors for a large range of its parameters, it is nonchaotic when its parameters satisfy one of seven conditions, which may help us understand the complex and rich dynamics of nonlinear laser models.

Degenerate soliton, breather and localized solutions for a nonlinear Schrödinger and Maxwell–Bloch system

In this paper, we aim to derive the degenerate soliton, breather and localized solutions for a nonlinear Schrödinger and Maxwell–Bloch system, which is governed by optical pulse propagation in an erbium-doped fiber. Based on the above solutions obtained, the degenerate bright and dark solitons (breathers) are observed and collision features between breathers and rogue waves are analyzed. Though Darboux-dressing transformation is an efficient method for deriving the localized solutions, by recalling and modifying the generalized Darboux transformation in this manuscript, we trust to find a novel and new way to derive the localized solutions. In detail, for complex field envelope q and polarization p, collisions between bright breathers and rogue waves are analyzed with corresponding parameters in the solutions. Meanwhile, for population inversion η, collisions between dark breathers and rogue waves are observed.

Homoclinic orbits and Jacobi stability on the orbits of Maxwell–Bloch system

In this paper, we analytically and geometrically investigate the complexity of Maxwell–Bloch system by giving new insight. In the first place, the existence of homoclinic orbits is rigorously proved by means of the generalized Melnikov method. More precisely, for 6a−2b>c and d>0, it is certified analytically that Maxwell–Bloch system has two nontransverse homoclinic orbits. Secondly, Jacobi stability on the orbits of Maxwell–Bloch system is examined in view point of Kosambi–Cartan–Chern theory (KCC-theory). In other words, in the light of the deviation curvature tensor of the five corresponding invariant associated to the reformulated Maxwell–Bloch system, we further proved that Jacobi stability of all equilibria under appropriate parameters. Moreover, the deviation vector, as well as the curvature of the deviation vector near equilibrium points, is focused to interpret the chaotic behavior of Maxwell–Bloch system in Finsler geometry.

Dynamics at Infinity and Existence of Singularly Degenerate Heteroclinic Cycles in Maxwell–Bloch System

Abstract In this paper, global dynamics of the Maxwell–Bloch system is discussed. First, the complete description of its dynamic behavior on the sphere at infinity is presented by using the Poincaré compactification in R3. Second, the existence of singularly degenerate heteroclinic cycles is investigated. It is proved that for a suitable choice of the parameters, there is an infinite set of singularly degenerate heteroclinic cycles in Maxwell–Bloch system. Specially, the chaotic attractors are found nearby singularly degenerate heteroclinic cycles in Maxwell–Bloch system by combining theoretical and numerical analyses for a special parameter value. It is hoped that these theoretical and numerical value results are given a contribution in an understanding of the physical essence for chaos in the Maxwell–Bloch system.

Acceleration techniques for semiclassical Maxwell–Bloch systems: An application to discrete quantum dot ensembles

The solution to Maxwell–Bloch systems using an integral-equation-based framework has proven effective at capturing collective features of laser-driven and radiation-coupled quantum dots, such as light localization and modifications of Rabi oscillations. Importantly, it enables observation of the dynamics of each quantum dot in large ensembles in a rigorous, error-controlled, and self-consistent way without resorting to spatial averaging. Indeed, this approach has demonstrated convergence in ensembles containing up to 104 interacting quantum dots (Glosser et al., 2017). Scaling beyond 104 quantum dots tests the limit of computational horsepower, however, due to the O(NtNs2) scaling (where Nt and Ns denote the number of temporal and spatial degrees of freedom). In this work, we present an algorithm that reduces the cost of analysis to O(NtNslog2Ns). While the foundations of this approach rely on well-known particle–particle/particle–mesh and adaptive integral methods, we add refinements specific to transient systems and systems with multiple spatial and temporal derivatives. Accordingly, we offer numerical results that validate the accuracy, effectiveness and utility of this approach in analyzing the dynamics of large ensembles of quantum dots.

Global dynamics of the Maxwell–Bloch system with invariant algebraic surfaces

In this paper by using the Poincaré compactification in we make a global analysis of the Maxwell–Bloch system with , b, c and . We give the complete description of its dynamics on the sphere at infinity. For some values of the parameters, this system has first integrals and invariant algebraic surfaces. For these sets, we provide the global phase portraits of the Maxwell–Bloch system in the Poincaré ball (i.e. in the compactification of with the sphere at infinity).

Darboux transformation and exact multisolitons for a matrix coupled dispersionless system

We present a matrix coupled dispersionless (CD) system. A Lax pair for the matrix CD system is proposed and Darboux transformation is constructed on the solutions of the matrix CD system and the associated Lax pair. We express an N soliton formula for the solutions of the matrix CD system in terms of quasideterminants. By using properties of the quasideterminants, we obtain some exact solutions, including bright and dark-type solitons, rogue wave and breather solutions of the matrix CD system. Furthermore, it has been shown that the solutions of the matrix CD system are expressed in terms of solutions to the usual CD system, sine-Gordon equation and Maxwell–Bloch system.

Near Resonant Optical Absorption by a System Coupled with Two Laser Beams

The structure of a solution of the generalized Maxwell–Bloch system of equations describing the strongly pumped interacting two-level atoms is discussed. This structure is represented by means of the corresponding differential equations for each contributing process. The interaction between the processes is introduced through the interaction integral and is illustrated by the specific system of graphs. The method allows one to describe the quantum-field-induced long-range interaction prevailing over short-range collisions and causing the broadening, narrowing, and shifts of an absorption line shape. The description is given in terms of the interaction integrals which couple the collective atomic polarization and population inversion. The contributions from different effects are analyzed with the use of the additivity of the corresponding absorption/reemission rates.

Rogue wave solutions for the (2+1)-dimensional complex modified Korteweg–de Vries and Maxwell–Bloch system

In this paper, we will focus on the (2+1)-dimensional complex modified Korteweg–de Vries and Maxwell–Bloch (cmKdVMB) system. According to the correlative Lax pair, the N-fold generalized Darboux Transformations (DT) will be constructed. By virtue of the DT obtained, some rogue wave solutions will be derived from the nonvanishing continuous wave (cw) backgrounds. Meanwhile, the dynamic features of those solutions are graphically analyzed.